Section 4.5 Using Exponential Functions to Model Data
Exponential Model, Exponential Related, Approximately Exponentially Related Using Base Multiplier Property to Find a Model Definition An exponential model is an exponential function, or its graph, that describes the relationship between two quantities for an authentic situation. If all of the data points for a situation lie on an exponential curve, then we say that the independent and dependent variables are exponentially related. (Continues) Section 4.5 Lehmann, Intermediate Algebra, 4ed Slide 2
Finding an Equation of an Exponential Curve Using the Base Multiplier Property to Find Exponential Functions Definition Continued If no exponential curve contains all of the data points, but an exponential curve comes close to all of the data points (and perhaps contains some of them), then we say that the variables are approximately exponentially related. Section 4.5 Lehmann, Intermediate Algebra, 4ed Slide 3
Exponential Model, Exponential Related, Approximately Exponentially Related Using Base Multiplier Property to Find a Model Example Suppose that a peach has 3 million bacteria on it at noon on Monday and that one bacterium divides into two bacteria every hour, on average. Let B be the number of bacteria (in millions) on the peach at t hours after noon on Monday. Section 4.5 Lehmann, Intermediate Algebra, 4ed Slide 4
Exponential Model, Exponential Related, Approximately Exponentially Related Using Base Multiplier Property to Find a Model Example Continued 1. Find an equation of f Solution Create table with assumptions As t increases by 1, B changes by multiplying by 2 So, model is of the form f(t) = a(2) t Slide 5
Exponential Model, Exponential Related, Approximately Exponentially Related Using Base Multiplier Property to Find a Model Solution Continued B-intercept is (0, 3), so f(t) = 3(2) t Verify solution using graphing calculator Example Continued 2. Predict the number of bacteria on the peach at noon on Tuesday. Slide 6
Exponential Model, Exponential Related, Approximately Exponentially Related Using Base Multiplier Property to Find a Model Solution Use t = 24 to represent noon on Tuesday Substitute 24 for t According to the model there would be 50,331,648 million bacteria Omit million : 50,331,648,000,000 That s about 50 trillion bacteria at noon on Tuesday Slide 7
Using Base Multiplier Property to Find a Model Summary Exponential function y = ab t the y-intercept is (0, a) y is the quantity at time t a is the quantity at time t = 0 Example A person invests $5000 in an account that earns 6% interest compounded annually. 1. Let V = f(t) be the value of the account at t years after the money is invested. Find an equation of f. Slide 8
Using Base Multiplier Property to Find a Model Solution Each year the investment value is the previous value (100%) plus 6% of the previous value Each year the investment is 106% of the previous year As t increases by 1, the vale of V is multiplied by 1.06: b = 1.06 Value of account starts at $5000: a = 5000 f(t) = 5000(1.06) t Slide 9
Using Base Multiplier Property to Find a Model Example Continued 2. What will be the value after 10 years? Solution Substitute 10 for t Value will be $8954.24 in 10 years Slide 10
Definition If a quantity decays exponentially, the half-life is the amount of time it takes for that quantity to be reduces to half. Definition: Half-life Half- life Applications Slide 11
Example Definition: Half-life Half- life Applications The world s worst nuclear accident occurred in Chernobyl, Ukraine, on April 26, 1986. Immediately afterward, 28 people died from acute radiation sickness. So far, about 25,000 people have died from exposure to radiation, mostly due to the release of the radioactive element cesium-137 (Source: Medicine Worldwide). Cesium-137 has a half-life of 30 years. Let P = f (t) be the percent of the cesium-137 that remains at t years since 1986. Slide 12
Example Continued 1. Find an equation of f. Solution Half- life Applications Method 1 At t = 0, 100% of the cesium-137 is present At t = 30, ½(100) = 50 percent is present At t = 60, ½(½)(100) = 25 percent Situation can be modeled using exponential function Slide 13
Half- life Applications Solution Continued Each exponent in second column is equal to the value of t in the first column divided by 30 Equation of f is: f t ( ) t 1 = 100 2 30 Slide 14
Half- life Applications Solution Continued Use a graphing calculator table and graph to verify solution 1 1 t t 30 t 30 30 1 1 1 f ( t) = 100 100 100 = = 2 2 2 1 30 1. 0.977 can be written f t = 100 0.977 2 ( ) ( ) t Slide 15
Half- life Applications Solution Continued Method 2 We know the points (0, 100) and (30, 50) P-intercept is (0, 100), so a = 100 Substitute (30, 50) Slide 16
Solution Continued Half- life Applications Equation is the same f t ( ) t 1 = 100 2 30 Slide 17
Half- life Applications Example Continued 2. Describe the meaning of the base of f. Solution Base of f is 0.977 Each year 97.7% of the previous year s cesium-137 is present Or, cesium-137 decays by 2.3% per year Slide 18
Example Continued 3. What percent of the cesium-137 will remain in 2010? Solution 2010 1986 = 24 Substitute 24 for t Half- life Applications In 2010, about 57.2% of the cesium-137 will remain Slide 19
Meaning of the Base of an Exponential Function Summary Half- life Applications If f(t) = ab t, where a > 0, models a quantity at time t, then the percentage of change is constant. In particular, If b > 1, then the quantity grows exponentially at a rate of b 1 percent (in decimal form) per unit of time. If 0 < b < 1, then the quantity decays exponentially at a rate of 1 b percent (in decimal form) per unit of time. Slide 20
Finding a Model by Using Data Described in Words Example Sales of energy and nutrition bars have grown approximately exponentially from $0.2 billion in 1997 to $1.2 billion in 2004 (Source: Frost & Sullivan). Predict the sales in 2011. Solution Let s be the sales (in billions of dollars) of energy and nutrition bars Let t be the years since 1997 Slide 21
Finding a Model by Using Data Described in Words Solution Continued Create a table t and s are approx. exponential We want s = ab t S-intercept is (0, 0.2) Substitute (7, 1.2) and solve for b Slide 22
Finding a Model by Using Data Described in Words Solution Continued Substitute 1.292 for b 2011 1997 = 14, so substitute 14 for t Slide 23
Finding a Model by Using Data Described in Words Solution Continued Model predicts that sales will reach $7.22 billion in 2011 Verify work on graphing calculator Slide 24